Modern Hadron Spectroscopy : Challenges and Opportunities Adam - - PowerPoint PPT Presentation

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Lecture 1: Hadrons as laboratory for QCD:

• Introduction to QCD
• Bare vs effective effective quarks and gluons
• Phenomenology of Hadrons

Lecture 2: Phenomenology of hadron reactions

• Kinematics and observables
• Space time picture of Parton interactions and Regge phenomena
• Properties of reaction amplitudes

Lecture 3: Complex analysis Lecture 4: How to extract resonance information from the data

• Partial waves and resonance properties
• Amplitude analysis methods (spin complications)

Modern Hadron Spectroscopy : Challenges and Opportunities

Adam Szczepaniak, Indiana University/Jefferson Lab

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Probing QCD resonances (using physical states)

2

• When (color neutral) mesons and baryons a smashed, their quarks
• verlap, “stick together” to form resonances (quasi QCD eigenstates).

They are short lived and decay to lowest energy, asymptotic states (pions, K’s, proton,…)

• Resonances are fundamental to our understanding of QCD dynamics

since they appear beyond perturbation theory.

• (QCD) Resonances challenge QFT practitioners to develop all orders

calculations (still ways to go).

• (QCD) Resonance lead to extremely rich phenomenology (e.g. XYZ

states).

• In practice, one requires tools that relate asymptotic states before collision

to asymptotic states after collision that include flexible parametrization of microscopic dynamics. This is often referred to as amplitude analysis. The rest of these lectures will focus on this topic.

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Bound states/Resonances/Asymptotic states

3

 p2 2me − α r

• ψ(r) = Eψ(r)

α = αQED = 1 137

ψ(r) = e−ikr r − S(E, θ)eikr r

ψ(r) ∝ e−αmer

S(E, θ) = 1 + O(α)

Born approximation : lowest order perturbation on free motion Bound states: compact wave function contains interaction to all orders. Resonances: particles interact to all orders (like bound states) but eventually decay (connect with asymptotically free states). Their effect appears in the S-matrix

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Amplitude analyticity: it is much about complex functions 4

Bound states Asymptotic states Resonances Scattering Amplitude Scattering amplitude describes evolution between asymptotic states. The information related to formation about resonances is “hidden” in unphysical domains (sheets) of the kinematical variables. This “bump” is an indication of a “hidden”

• phenomenon. To uncover it one needs to

analytically continue outset the physical sheet

A(s + i✏) = Aphysical(s = real and above threshold)

s = E2

c.m

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Introduction to Scattering

5

• Time evolution pictures: Schrodinger, Heisenberg, Interaction

H = Hkin + V → H0 + V (t)

OI(t) = eiH0O(0)e−iH0t

|tiI = eiH0t|tiS

VI(t) = eiH0V e−iH0te−✏|t| H0,I(t) = H0

i d dt|tiI = VI(t)|tiI

V → V (t) = V e−✏|t|

• As t → ± ∞ interaction picture states evolve to eigenstates of Hkin, i.e. to free

particles

• At t=0 interactions picture states are solution of the full Hamiltonian

Interaction is switched on adiabatically at t=0

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S-matrix and T-matrix

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i d dt|tiI = VI(t)|tiI

|tiI = U(t, 1)|initiali Sfi = hf(t = +1)|i(t = 1i = hf, (out)|i, (in)i

= hf|U(+1, 1)|ii

U(+∞, −∞) = P exp ✓ −i Z +∞

−∞

dtVI(t) ◆

Evolution operator

• S-matrix
• T-matrix

T = V + V G0V + · · ·

= I − 2πiδ(Ef − Ei)T

G0 = 1 E − H0

E = Ei = Ef

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T matrix : Example

7

Example Method 1: In coordinate space Method 2: Lippmann-Schwinger (see above) Nonrelativistic particle scarring in external potential

• It has ∞ number of zeros (this is related to ∞ number of poles when calculated to all
• rders)

T = V + V G0V + · · ·

V = λ 2µa2 δ(r − a)

dimλ = −1

H = p2 2µ + V

r V(r) a ε 1/ε

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Solution

8

From method 1

f(k) = h −λ sin2(ka)

(ka)2

i h 1 + λ

a sin(ka) cos(ka) ka

i − ik h −λ sin2(ka)

(ka)2

i

f(k) = K(E) 1 − iK(E)k = 1 K−1(E) − ik

∞ of zeros ∞ of zeros → Poles

E = k2/2µ

= P (k)

Q(k)

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Solution

9

From method 2

f(k) = −λ sin2(ka)

(ka)2

1 − 1

π

R 1 dE0k0 λ sin2(k0a)

(k0a)2

E0E(k)

k0 = k(E0) = √2µE0

K(E) =

λ sin2(ka)

(ka)2

1 1

π <

R ···

(∞) zeros of K !

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Analyticity

10

K(k = kR + ikI) → const. + O(e−2kIa) =

1 ik + O(e−2kIa)

f(k) =

K(E) 1−iK(E)k = O(e+2ikIa) Essential singularity at infinity in the physical sheet ! “Conspiracy” between zeros and poles !!! E.g. ∞ number of zeros of K(s) are “fixed” by geometry of the sphere (“dynamics”) and this specific “physics” fixes all the poles. In more general case (no fixed scattering radius) correlation between zeroes and poles persist”, an infinite number poles requires infinite number of zeroes (and vice versa)

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S-matrix properties (in relativistic theory)

22

X

f

Pfi = 1

2ImTft = X

n

2πδ(Ei − En)T ∗

fnTni

• Related to transition probability
• Conservation of Probability = Unitarity

Pfi = |hf|S|ii|2 = hi|S†|fihf|S|ii S†S = I

• Lorentz symmetry: T is a product of Lorentz scalars and covariant factors

representing wave functions of external states, e.g for

• Crossing symmetry: the same scalar functions describe all process related by

permutation of legs between initial and final states (only the wave function change)

• Analyticity: The scalar functions are analytical functions of invariants

¯ u(p1, λ1)[A(s, t) + (k1 + k2)µγµB(s, t)]u(p2, λ2) ¯ v(p1, µ1)[A(s, t) + (k1 + k2)µγµB(s, t)]u(p2, µ2)

π(k1) + N(p1, λ1) → π(k2) + N(p2, λ2)

π(k1) + π(−k2) → ¯ N(−p1, µ1) + N(p2, µ2)

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Lorentz symmetry

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p2 p3 p4 p1

b a c d

u = (p1 − p4)2 < 0 t = (p1 − p3)2 < 0 s = (p1 + p2)2 > (ma + mb)2

= (E1,cm + E2,cm)2

t = m2

1 + m2 2 − 2E1,cmE2,cm + 2|p1,cm||p2,cm|zs

u = m2

1 + m2 4 − 2E1,cmE4,cm − 2|p1,cm||p4,cm|zs

s + t + u = X

i

m2

i

hp0, β|p, αi = 2E(p)δ(pf pi)δα,β

2πδ(Ef Ei)iT = hc, d|(S 1)|a, bi

T = (2π)3δ(pf − pi)A(s, t, u)

N-to-M scattering depends on 4(N+M)-4-10 = 3(N+M)-10 invariants e.g for 2-to-2: 2 invariants related to the c.m. energy and scattering angle Dimensions r.h.s has dim = -4 A(s,t,u) is a scalar function of mass dimension =0

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Question

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How many independent variables describe

• Decay proces A → a + b +c
• Three particle production A +B → a + b + c

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Helicity amplitudes

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hp3, λ3; p4, λ4|A|p1, λ1; p2, λ2i = Aλ1,λ2,λ3,λ4(s, t, u)

~ S · ~ p |~ p| |p, i = |p, i

Sz|p, miz = m|p, miz

|p, i = R(ˆ p)Λ(|~ p|ˆ z 0)|0, miz

|p, miz = Λ(~ p 0)|0, miz

|p, λiz =

S

X

m=−S

|p, mizDS

m,λ(ˆ

p) Aλ1,λ2,λ3,λ4(s, t, u) = ηA−λ1,−λ2,−λ3,−λ4(s, t, u)

We work in the c.m. frame Helicity states vs canonical spin states: Exercise show this: Parity

• Even though this looks non relativistic it is relativistic. Notion of LS amplitudes,

LS vs. helicity relations are relativistic

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Question

26

How many independent scalar functions describe J/ψ → π+ π- π0 Ɣ p-> π0 p

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Crossing symmetry

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1 2 3 4 a(p1) + b(p2) → c(p3) + d(p4) 1 2 3 4 1 2 3 4 _ _ _ a(p1) + c(p3) → b(p2) + d(p4) _ _ _ _ _ a(p1) + d(p4) → c(p3) + b(p2) _ _ _ _

¯ pi = −pi = (−~ pi, −Ei)

u = (p1 − p4)2 s = (p1 + p2)2 t = (p1 − p3)2

Ec.m Cos(θ) Cos(θ)

s = (p1 − p¯

2)2

t = (p1 + p¯

3)2

u = (p1 − p4)2 u = (p1 + p¯

4)2

s = (p1 − p¯

2)2

t = (p1 − p3)2

s t u

A(s)

λ1,···(s + i✏, t, u) →

X

λ0

1,···

[DS1

λ1,λ0

1 · · · ]A(t)

λ0

1,···(s, t + i✏, u) → · · ·

• The iε is important. Function values at, e.g. s + iε vs s - iε are different !

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Crossing Symmetry : Decays

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1 2 3 4 1 2 3 4 _ a(p1) + b(p2) → c(p3) + d(p4) a(p1) → b(p2) + c(p3) + d(p4) _ _

M1 > m2 + m3 + m4 A(s, t, u) → A(M 2

1 + i✏, s + i✏, t + i✏, u + i✏)

• In decay kinematics, the decaying mass becomes a dynamical variable, (iε

important)

• Crossing from one kinematical region (e.g. s-channel) to another (e.g. t-channel)

requires taking the corresponding variables off the real axis and to the complex plane : analytical continuation.

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Analyticity

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Feynman diagrams

p2 p3 p4 p1

b a c d

k1 p2-k1

1 m2

q − (p2 − k1)2

1 k2 A(p1, · · · ) ∝ Z [Πjd4kj] polynomial in kj (m2

q − (pi − kj)2 − i✏)((ki − kj)2 − i✏) · · ·

m2 − p2 = [m2 + p2] − (p0)2 m2 − p2 = 0 → p0 = ±(m2 + p2)1/2

Im " 1 p m2 + p2 ⌥ i✏ p0 # = ±⇡(p0 p m2 + p2)

• Integrand becomes singular when

intermediate states go on shell.

• Thresholds for producing physical

intermediate are the only reason why amplitudes are singular.

• Production of intermediate states is related to
• unitarity. Thus we expect unitarity to

determine singularities of the amplitudes. On the role of iε

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Analyticity and Causality

30

Dispersion relations

source emits a signal at t=0 causality: receiver receives at t>0 and not at t<0 amplitude of the signal consider the Fourier transform (E → energy) and extend definition to complex plane E → z, then f(z) is holomorphic for Im E > 0

f(t) ∝ θ(t) f(E) ≡ Z dteiEtf(t)

Causality: The outgoing wave cannot appear before the incoming one. Causality determines analytical properties of the scattering amplitude as function on energy/ momenta/scattering angle. The specific from of these conditions depend on the type of interactions and kinematics (e.g. relativistic vs non relativistic)

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momentum vs energy planes

31

f ∗(k) = f(−k∗) f ∗(E) = f(E∗)

E = k2 2µ

k = p 2µE

k E

The function is analytical in the whole E-plane not only the upper half

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How unitarity constrains singularities

32

2ImTft = X

n

2πδ(Ei − En)T ∗

fnTni

A(s + i✏) = Aphysical(s = real and above threshold)

• Unitarity “operates” in the physical domain, i.e. s real and above threshold

and |Cos(θ)|<1. This domain is the boundary of the complex plane where analytical amplitude are defined

sign fixed by “arrow of time V(t) = V exp(-t |ε|)

• The difference (discontinuity) A(s + iε) - A(s - iε) ≠ 0 (cf. Feynman diagrams),

comes from particle production this we expect it being determined by unitarity.

• Cauchy theorem : singularities determine the amplitude !!!

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Relativistic S-matrix fundamentals

33

Causality: Determines domain of analyticity of reaction amplitudes as function of kinematical variables. Unitarity: Determines singularities. Crossing: Dynamical relation, aka reaction amplitudes in the exchange channel (forces) are analogous to amplitude in the direct channel (resonance)

These defined the Bootstrap program of the 60’s. It is equivalent to non- relativistic QM, but not to QFT, i.e. “bootstrap equations” do not have unique

• solutions. For example it failed to reproduce the QCD resonance spectrum,

which needs ”external parameters”. (aka. K-matrix poles, CDD -poles, etc.)

P1 P5 P3

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How unitarity constrains singularities: simple example

34

2ImTft = X

n

2πδ(Ei − En)T ∗

fnTni

ImA(s, t) = ρ(s) 16π Z dΩ 4π A(s, cos θ1)A∗(s, cos θ2)

1 2 3 4 1 2 3 4 5 6

= 1 2 X

5,6

Im A(s, t) = 16π

∞

X

l=0

(2l + 1)fl(s)Pl(cos θ) Imfl(s) = ρ(s)|fl(s)|2

Consider elastic scattering of spineless particles

ρ(s) = 2kcm(s)/√s

At fixed s, this is a complicated, integral relation w.r.t momentum transfer, t It is simplified (diagonalized) by expanding A(s,t) in partial waves

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How unitarity constrains singularities

35

fl(s) = 1 32π Z 1

−1

d cos θPl(cos θ)A(s, t(s, cos θ))

→ Reflection theorem (Calculus 101): fl(s*) = fl(s*) Properties of the partial wave, fl(s) (for fixed l as function of s):

• fl(s) is real for s below threshold
• Im fl(s) is finite above threshold.
• fl(s) is complex for diffidently negative s
• fl(s) is analytical (since A(s,t) is)

fl(s+iε) fl(s-iε) Threshold s=(m1+m2)2

• Even though fl(s) has physical meaning for s

real and above threshold, there is a unique function in the complex plane which reduces to fl(s) on the real axis (+iε).

• Furthermore, unitarity which is a condition

for physical s-values, becomes a restriction

• n the complex function, fl(s).

1 2i[fl(s + i✏) − fl(s − i✏)] = ⇢(s)fl(s + i✏)fl(s − i✏)

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Second sheet

36

f(s + i✏) = f(s − i✏) 1 − 2i⇢(s)f(s − i✏) fII(s) = f(s) 1 − 2iρ(s)f(s) fII(s − i✏) = f(s + i✏) f(s) = 1 2iρ(s)

Singularity = Resonance at complex s when Define for Im s < 0 This is analytical continuation of f(s) below the real axis

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Breit-Wigner Formula

37

BW(s) = 1 m2

r − s − imrΓ(s)

Γ(s) = kcm(s)γ(s)

Threshold factor “rest”

kcm ∼ √s − sth |BW|2 Res Ims BW(s) ∼ 1 2 − s − 0.8i√s − 1 Ims |BW|2 Res BW(s) ∼ 1 2 − s + 0.8√1 − s

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Kinematical vs Dynamical Singularities

38

λ = λ1 − λ2 λ0 = λ3 − λ4 M = max(|λ|, |λ0|) λ1 λ2 λ4 λ3 Aλi(s, t) = 16π

M

X

J=−M

(2J + 1)f J

λi(s)dJ λ,λ0(θ)

f J

λi(s) =

1 32π Z 1

−1

dzsAλi(s, t(s, θ))dJ

λ,λ0(θ)

For particles with spin

• Wigner d-functions lead to kinematical singularities
• Threshold (barrier factors) originate from kinematical factors in relation

between t and cos(θ) (through dependence of Aλ on t)

• Unequal masses give lead to “daughter poles”
• Dynamical singularities : from dynamical (unitary cuts) in A(s,t).

from M.Ostrick

∆(1232)3/2+

N(1520)3/2−

N(1680)5/2+

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Phenomenology of hadron interaction

39

σa+b→a+b ∝ Z dt s2 |A(s, t)|2 σa+b→X ∝ ImA(s, 0) s

from unitarily Resonance scattering

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Resonance Scattering

40

dσ dt ∝ |A(s, t)|2 s2

from M.Ostrick

Angular distribution: a few “wiggles” more pronounced forward/backward peaks as energy increases

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Resonance scattering

41

+ + ...

A(s, t) = X

l

(2l + 1)fl(s)Pl(zs(t))

A(s, t) ∼ PlR(zs(t)) s − sR

• If QCD was confining resonance would appear at all energy and angular momenta

(infinitely rising Regge trajectories).

• String/flux tube breaking leads to screening of color charge and resonance seem to

appear with finite angular momentum.

• For lmax ~ 5 and nteraction range r0 ~0.5fm this gives plab <~ 10/fm ~ 2GeV,

[or W ~ (2 Plab mp )1/2 ~ 2GeV ]

• For resonance scattering

Multiple quark/gluon exchanges

=

→

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Scattering at High energies

42

dσ dt (s) = 1 s2 |A(s, t)|2

σa+b→X = 1 sImAab→ab(s, 0)

Smooth behavior constant or power low fall off Smooth fall of with t and forward/backward peking

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Scattering at high energies

43

• s-dependence:
• many intermediate particles can be produced, unitarity becomes

complicated and less useful.

• t-dependence:
• high partial waves become important, several Legendre functions are

needed.

• There is universality in both s and t-dependencies: smooth (constant or falling

s-dependence), and forward/(backward) peaking in t. The universality hints into importance of t/(u) channel singularities.

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From t-channel to s-channel (high energy forward scattering)

44

t s u s=4m2 t=4m2 u=4m2 a+b->c+d s-channel a+c->b+d t-channel

• a+d->c+b

u-channel

• As s increase and t is fixed the

t-channel resonances (or singularities) stay close relative to s and u channel resonances

s increases t is fixed

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From u-channel to s-channel (high energy backward scattering)

45

t s u s=4m2 t=4m2 u=4m2 a+b->c+d s-channel a+c->b+d t-channel

• a+d->c+b

u-channel

• s increases

u is fixed

As s increase and u is fixed the u-channel resonances (or singularities) stay close relative to s and t channel resonances

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Regge phenomena

46

A(s, t) = X

l

(2l + 1)fl(s)Pl(zs(t))

for s ~ sR A(s, t) ∼ PlR(zs(t))

s − sR

Low energies: resonance dominance ...looks like resonance in t-channel: hint explore unitarity in t-channel

to rigorously establish the large-s behavior one needs to make sense out of the divergent sum. (Gribov-Froissart projection + Somerfeld-Watson transformation)

resonance poles in fl(t) at t=tR can be considered as poles in l: tR=t(l) --> l=α(tR)

A(s, t) = X

l

(2l + 1)fl(t)Pl(zt(s)) A(s, t) ∼ sl = sα(t)

thus for large s = + ... + + ... + s t

zs = cos θs

in principle A(s,t) determined from s- channel unitarity (s-channel dispersion relations) but there are many intervening channels... How to extend to high energies t s

zt = cos θt

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Example of analytical continuation

47

1 l − α = Z ∞ dxe−x(l−α) J(z) = Z ∞ dx  exα 1 + ze−x

• = zα

Z z dy yα+1(1 + y)

to obtain

y = ze−x

For example, assume i.e. it has a pole (resonance) where α(t)=l z → ∞ for large z =z(s) ~ s

J(z) = − zαπ sin πα + zα Z ∞

z

dy yα+1(1 + y) → − zαπ sin πα

provides analytical continuation for α>0

A(s, t) = X

l

(2l + 1)fl(t)Pl(zt)

The series converges for |zt|<1 (cosine of scattering angle in the t-channel), i.e. in the t-channel physical region. We want to know A(s,t) for in the s-channel physical region, in particular for large s, with corresponds to |zt| >> 1.

s = −t − 4m2 2 (1 − zt)

fl(t) = 1 l − α(t)

for α<0 and |zt| < 1 use

A(s, t) ∼ J(zt) = X

l

zl

t

l − α(t) In general use Sommefeld-Watson transformation to sum a series

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Pomeron vs Reggeons

48

s-channel: multi-particle production t-channel: collection of resonances: “Regge” exchanges

fl(t) = r(t) l − α(t)

<--

A(s, t) ∝ r(t)sα(t)

γp → X

Rightmost singularity in l-plane dominates large-s limit of the amplitude and forward cross section (it has vacuum quantum numbers: Pomeronα(0) = 1 + ε) (exchange of non-vacuum q.n. falls with energy)

σtot ∼ s = s0.08

σel ∼ 1 b s2α(0)−2

A(s, t ∼ 0) ∼ isα(0) ∼ sσtot

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Comparing with Experiment

49

resonance region Ecm = s1/2 < 2.5 GeV multi-particle production total cross section slowly rises with s

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Growing Radius, partons, saturation,…

50

long lived fluctuations finite <x>

∆E ∼ µ2

⊥

x(1 − x)pz pz → ∞ p = 0 (1 − x)pz

interaction when commensurate momenta

hxihni = pz µ hni ⇠ log(s)

random walk in transverse space

hr⊥i ⇠ r hni 1 µ⊥ ⇠ log1/2(s)

large-s behavior universal (Pomeron = vacuum pole, universal mid-rapidity) Where does to parton model come from

A(s, r⊥) ∼ Z d2k⊥eik⊥r⊥eα(−k2

⊥) log s ∼

1 log(s)e−r2

⊥/ log(s)

... and in space-time assuming Pomeron α(0)=1 hadron swells

(slow moving hadron,vacuum,etc)

g2 s X

n

βn−1(t) (n − 1)! logn−1 s → sα(−k2

⊥)

α(t) = −1 + β(t)

(fast moving, hadron, parton,etc)

adding correlated partons is beneficial (expansion not in g2 but in g2 log s ) it takes “a long time” to develop a low-x parton out of a fast one

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Summary of Lecture 2

51

S-matrix principles : Crossing symmetry, Analyticity, Unitarity provide important constraints/insights into reaction dynamics. For example: low energy scaring is dominated by a few direct channel partial waves, resonance appear as poles on the IInd sheet with widths constrained by unitarity, large-s scattering is given by t/u channel exchanges, etc. In QCD resonances are not predicted by exchange forces (Bootstrap idea), they have to be “inserted by hand”.